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Theorem vtocl2d 29442
Description: Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.)
Hypotheses
Ref Expression
vtocl2d.a (𝜑𝐴𝑉)
vtocl2d.b (𝜑𝐵𝑊)
vtocl2d.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))
vtocl2d.3 (𝜑𝜓)
Assertion
Ref Expression
vtocl2d (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem vtocl2d
StepHypRef Expression
1 vtocl2d.b . . 3 (𝜑𝐵𝑊)
2 vtocl2d.a . . 3 (𝜑𝐴𝑉)
3 nfcv 2793 . . . 4 𝑦𝐵
4 nfcv 2793 . . . 4 𝑥𝐵
5 nfcv 2793 . . . 4 𝑥𝐴
6 nfv 1883 . . . . 5 𝑦𝜑
7 nfsbc1v 3488 . . . . 5 𝑦[𝐵 / 𝑦]𝜓
86, 7nfim 1865 . . . 4 𝑦(𝜑[𝐵 / 𝑦]𝜓)
9 nfv 1883 . . . 4 𝑥(𝜑𝜒)
10 sbceq1a 3479 . . . . 5 (𝑦 = 𝐵 → (𝜓[𝐵 / 𝑦]𝜓))
1110imbi2d 329 . . . 4 (𝑦 = 𝐵 → ((𝜑𝜓) ↔ (𝜑[𝐵 / 𝑦]𝜓)))
12 sbceq1a 3479 . . . . . 6 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜓[𝐴 / 𝑥][𝐵 / 𝑦]𝜓))
13 nfv 1883 . . . . . . . 8 𝑥𝜒
14 nfv 1883 . . . . . . . 8 𝑦𝜒
15 nfv 1883 . . . . . . . 8 𝑥 𝐵𝑊
16 vtocl2d.1 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))
1713, 14, 15, 16sbc2iegf 3537 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
182, 1, 17syl2anc 694 . . . . . 6 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
1912, 18sylan9bb 736 . . . . 5 ((𝑥 = 𝐴𝜑) → ([𝐵 / 𝑦]𝜓𝜒))
2019pm5.74da 723 . . . 4 (𝑥 = 𝐴 → ((𝜑[𝐵 / 𝑦]𝜓) ↔ (𝜑𝜒)))
21 vtocl2d.3 . . . 4 (𝜑𝜓)
223, 4, 5, 8, 9, 11, 20, 21vtocl2gf 3299 . . 3 ((𝐵𝑊𝐴𝑉) → (𝜑𝜒))
231, 2, 22syl2anc 694 . 2 (𝜑 → (𝜑𝜒))
2423pm2.43i 52 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469
This theorem is referenced by:  submateq  30003
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